geometric series
http://plus.maths.org/content/taxonomy/term/337
enWinning odds
http://plus.maths.org/content/os/issue55/features/nishiyama/index
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Yutaka Nishiyama and Steve Humble </div>
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<img class="imagefield imagefield-field_abs_img" width="100" height="100" alt="" src="http://plus.maths.org/content/sites/plus.maths.org/files/abstractpics/5/12%20Jul%202010%20-%2015%3A49/icon-6.jpg?1278946144" /> </div>
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When you flip a coin we assume it has equal chance of coming up head or tails, so any coin flipping game should be a fair one. But <b>Yutaka Nishiyama</b> and <b>Steve Humble</b> can give you the winning advantage. </div>
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<br clear="all"><p>One day I received an email from my co-author, Steve Humble. In some excitement, he told me that a magician named Derren Brown was introducing an interesting game on <a href="http://derrenbrown.channel4.com/derren-brown-penney-ante-game.shtml">television</a>. I was a little dubious upon hearing the word "magician", but after close examination I realised that the game had a mathematical background and was an interesting exercise in probability. </p><p><a href="http://plus.maths.org/content/os/issue55/features/nishiyama/index" target="_blank">read more</a></p>http://plus.maths.org/content/os/issue55/features/nishiyama/index#comments55card gamescoin gamesgeometric seriesleading numbersoddsPenney anteprobabilityMon, 12 Jul 2010 15:04:32 +0000mf3445224 at http://plus.maths.org/contentAn infinite series of surprises
http://plus.maths.org/content/infinite-series-surprises
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C. J. Sangwin </div>
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<img class="imagefield imagefield-field_abs_img" width="130" height="130" alt="" src="http://plus.maths.org/content/sites/plus.maths.org/files/issue19/features/infseries/icon.jpg?1007164800" /> </div>
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Infinite series occupy a central and important place in mathematics. <b>C. J. Sangwin</b> shows us how eighteenth-century mathematician Leonhard Euler solved one of the foremost infinite series problems of his day. </div>
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<div class="pub_date">Mar 2002</div>
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<h3>Introduction</h3>
[maths]An infinite sum of the form \setcounter{equation}{0} \begin{equation} a_1 + a_2 + a_3 + \cdots = \sum_{k=1}^\infty a_k, \end{equation} is known as an infinite series. Such series appear in many areas of modern mathematics. Much of this topic was developed during the seventeenth century. Leonhard Euler continued this study and in the process solved many important problems.<p><a href="http://plus.maths.org/content/infinite-series-surprises" target="_blank">read more</a></p>http://plus.maths.org/content/infinite-series-surprises#comments19convergencedivergenceEuler's solution to the Basel problemgeometric seriesharmonic seriesinfinite seriesintegral testpower seriesSat, 01 Dec 2001 00:00:00 +0000plusadmin2202 at http://plus.maths.org/contentMathematical mysteries: Zeno's Paradoxes
http://plus.maths.org/content/mathematical-mysteries-zenos-paradoxes
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Rachel Thomas </div>
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<img class="imagefield imagefield-field_abs_img" width="100" height="100" alt="" src="http://plus.maths.org/content/sites/plus.maths.org/files/abstractpics/5/9_jun_2011_-_1545/tort1.jpg?1307630744" /> </div>
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<p>The paradoxes of the philosopher Zeno, born approximately 490 BC in southern Italy, have puzzled mathematicians, scientists and philosophers for millennia. Although none of his work survives today, over 40 paradoxes are attributed to him which appeared in a book he wrote as a defense of the philosophies of his teacher Parmenides.</p>
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<div class="pub_date">Nov 2001</div>
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<p>The paradoxes of the philosopher <a href="http://www-groups.dcs.st-andrews.ac.uk/~history/Mathematicians/Zeno_of_Elea.html">Zeno</a>, born approximately 490 BC in southern Italy, have puzzled mathematicians, scientists and philosophers for millennia. Although none of his work survives today, over 40 paradoxes are attributed to him which appeared in a book he wrote as a defense of the
philosophies of his teacher Parmenides.<p><a href="http://plus.maths.org/content/mathematical-mysteries-zenos-paradoxes" target="_blank">read more</a></p>http://plus.maths.org/content/mathematical-mysteries-zenos-paradoxes#comments17Achilles ParadoxArrow Paradoxconvergencegeometric serieslimitMathematical mysteriesrelativityworldlineZeno's paradoxesFri, 01 Dec 2000 00:00:00 +0000plusadmin4753 at http://plus.maths.org/contentIn perfect harmony
http://plus.maths.org/content/perfect-harmony
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John Webb </div>
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<img class="imagefield imagefield-field_abs_img" width="130" height="130" alt="" src="http://plus.maths.org/content/sites/plus.maths.org/files/issue12/features/harmonic/icon.jpg?967762800" /> </div>
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The harmonic series is far less widely known than the arithmetic and geometric series. However, it is linked to a good deal of fascinating mathematics, some challenging Olympiad problems, several surprising applications, and even a famous unsolved problem. <b>John Webb</b> applies some divergent thinking, taking in the weather, traffic flow and card shuffling along the way. </div>
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<div class="pub_date">September 2000</div>
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<h2>Introduction</h2>
<p>Two elementary series are studied in school mathematics:</p>
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<li>arithmetic series, such as <img src="/MI/2e864830e5e62b8fc35f2ecdbfbc9d6a/images/img-0001.png" alt="$1+2+3+\dots +n$" style="vertical-align:-2px;
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<p>and</p><p><a href="http://plus.maths.org/content/perfect-harmony" target="_blank">read more</a></p>http://plus.maths.org/content/perfect-harmony#comments12arithmetic seriesconvergencedivergencegeometric seriesharmonic serieslogarithmThu, 31 Aug 2000 23:00:00 +0000plusadmin2172 at http://plus.maths.org/contentHave we caught your interest?
http://plus.maths.org/content/have-we-caught-your-interest
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John H. Webb </div>
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<img class="imagefield imagefield-field_abs_img" width="130" height="130" alt="" src="http://plus.maths.org/content/sites/plus.maths.org/files/issue11/features/compound/icon.jpg?959814000" /> </div>
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Those who understand compound interest are destined to collect it. Those who don't are doomed to pay it - or so says a well-known source of financial advice. But what is compound interest, and why is it so important? <strong>John H. Webb</strong> explains. </div>
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<div class="pub_date">June 2000</div>
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<h2>Introduction</h2>
<center><em>"Neither a borrower nor a lender be."</em></center>
<p>Today, nobody heeds the advice of Polonius to his student son Laertes. Everybody borrows and lends all the time.</p><p><a href="http://plus.maths.org/content/have-we-caught-your-interest" target="_blank">read more</a></p>http://plus.maths.org/content/have-we-caught-your-interest#comments11compound interestdifferential equationfuture valuegeometric serieslogarithmMaclaurin seriespresent valuerule of 70Wed, 31 May 2000 23:00:00 +0000plusadmin2166 at http://plus.maths.org/content